`Learning.IsAlgEnvSeq.law_pullCount_sumRewards_unique'`🔗

Lemma

No docstring.

theorem

Learning.IsAlgEnvSeq.law_pullCount_sumRewards_unique'.{u_1, u_2, u_3}
  {𝓐 : Type u_1} {Ω : Type u_2} {Ω' : Type u_3} [DecidableEq 𝓐]
  {m𝓐 : MeasurableSpace 𝓐} {mΩ : MeasurableSpace Ω}
  {mΩ' : MeasurableSpace Ω'} {P : MeasureTheory.Measure Ω}
  [MeasureTheory.IsProbabilityMeasure P] {P' : MeasureTheory.Measure Ω'}
  [MeasureTheory.IsProbabilityMeasure P'] {alg : Algorithm 𝓐 ℝ}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ}
  [ProbabilityTheory.IsMarkovKernel ν] {A : ℕ → Ω → 𝓐} {R : ℕ → Ω → ℝ}
  {A₂ : ℕ → Ω' → 𝓐} {R₂ : ℕ → Ω' → ℝ} {n : ℕ}
  [MeasurableSingletonClass 𝓐]
  (h1 : IsAlgEnvSeq A R alg (stationaryEnv ν) P)
  (h2 : IsAlgEnvSeq A₂ R₂ alg (stationaryEnv ν) P') :
  ProbabilityTheory.IdentDistrib
    (fun ω a => (pullCount A a n ω, sumRewards A R a n ω))
    (fun ω a => (pullCount A₂ a n ω, sumRewards A₂ R₂ a n ω)) P P'
Learning.IsAlgEnvSeq.law_pullCount_sumRewards_unique'.{u_1,
    u_2, u_3}
  {𝓐 : Type u_1} {Ω : Type u_2}
  {Ω' : Type u_3} [DecidableEq 𝓐]
  {m𝓐 : MeasurableSpace 𝓐}
  {mΩ : MeasurableSpace Ω}
  {mΩ' : MeasurableSpace Ω'}
  {P : MeasureTheory.Measure Ω}
  [MeasureTheory.IsProbabilityMeasure P]
  {P' : MeasureTheory.Measure Ω'}
  [MeasureTheory.IsProbabilityMeasure P']
  {alg : Algorithm 𝓐 ℝ}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ}
  [ProbabilityTheory.IsMarkovKernel ν]
  {A : ℕ → Ω → 𝓐} {R : ℕ → Ω → ℝ}
  {A₂ : ℕ → Ω' → 𝓐} {R₂ : ℕ → Ω' → ℝ}
  {n : ℕ} [MeasurableSingletonClass 𝓐]
  (h1 :
    IsAlgEnvSeq A R alg (stationaryEnv ν)
      P)
  (h2 :
    IsAlgEnvSeq A₂ R₂ alg
      (stationaryEnv ν) P') :
  ProbabilityTheory.IdentDistrib
    (fun ω a =>
      (pullCount A a n ω,
        sumRewards A R a n ω))
    (fun ω a =>
      (pullCount A₂ a n ω,
        sumRewards A₂ R₂ a n ω))
    P P'

Code

lemma _root_.Learning.IsAlgEnvSeq.law_pullCount_sumRewards_unique' [MeasurableSingletonClass 𝓐]
    (h1 : IsAlgEnvSeq A R alg (stationaryEnv ν) P)
    (h2 : IsAlgEnvSeq A₂ R₂ alg (stationaryEnv ν) P') :
    IdentDistrib (fun ω a ↦ (pullCount A a n ω, sumRewards A R a n ω))
      (fun ω a ↦ (pullCount A₂ a n ω, sumRewards A₂ R₂ a n ω)) P P'

Proof

by
  have hA := h1.measurable_action
  have hR := h1.measurable_feedback
  have hA2 := h2.measurable_action
  have hR2 := h2.measurable_feedback
  constructor
  · refine Measurable.aemeasurable ?_
    rw [measurable_pi_iff]
    exact fun a ↦ Measurable.prod (by fun_prop) (measurable_sumRewards hA hR _ _)
  · refine Measurable.aemeasurable ?_
    rw [measurable_pi_iff]
    exact fun a ↦ Measurable.prod (by fun_prop) (measurable_sumRewards hA2 hR2 _ _)
  have h_unique := isAlgEnvSeq_unique h1 h2
  let f := fun (p : ℕ → 𝓐 × ℝ ) (a : 𝓐) ↦ (∑ i ∈ range n, if (p i).1 = a then 1 else 0,
    ∑ i ∈ range n, if (p i).1 = a then (p i).2 else 0)
  have hf : Measurable f := by
    rw [measurable_pi_iff]
    intro a
    refine Measurable.prod ?_ ?_
    · simp only [f]
      refine measurable_sum _ fun i hi ↦ Measurable.ite ?_ (by fun_prop) (by fun_prop)
      exact (measurableSet_singleton _).preimage (by fun_prop)
    · simp only [f]
      refine measurable_sum _ fun i hi ↦ Measurable.ite ?_ (by fun_prop) (by fun_prop)
      exact (measurableSet_singleton _).preimage (by fun_prop)
  have h_eq_comp : (fun ω a ↦ (pullCount A a n ω, sumRewards A R a n ω))
      = f ∘ (trajectory A R) := by
    ext ω a : 2
    rw [pullCount_eq_comp (R := R), sumRewards_eq_comp]
    grind
  have h_eq_comp2 : (fun ω a ↦ (pullCount A₂ a n ω, sumRewards A₂ R₂ a n ω))
      = f ∘ (trajectory A₂ R₂) := by
    ext ω a : 2
    rw [pullCount_eq_comp (R := R₂), sumRewards_eq_comp]
    grind
  rw [h_eq_comp, h_eq_comp2, ← Measure.map_map hf, h_unique, Measure.map_map hf,
    ← h_eq_comp2]
  · rw [measurable_pi_iff]
    exact fun n ↦ Measurable.prodMk (hA2 n) (hR2 n)
  · rw [measurable_pi_iff]
    exact fun n ↦ Measurable.prodMk (hA n) (hR n)

Actions: Source · Open Issue

Meaning last changed in v4.34.0-rc2-14-gf86702d (2026-08-25), the 5th recorded change.

Self-contained, with its dependencies inlined and proofs replaced by sorry: download the raw file · open it in the Lean web editor.

Dependency graph

Audit surface: 8 project declarations, 63 external constants

✓ Proved: no sorry anywhere in its closure

This is the tool's own reading of one build's recorded axioms, and it is not robust against an author who wants it to pass. Checking meant to be relied on should go through Comparator, which replays the proof through the kernel from an export against an explicit list of permitted axioms.

Learning.IsAlgEnvSeq.law_pullCount_sumRewards_unique'🔗

`Learning.IsAlgEnvSeq.law_pullCount_sumRewards_unique'`🔗